Mechanisms in wing-in-ground effect aerodynamics

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Mechanisms in Wing-In-Ground Effect

Aerodynamics

M arvin A lan Jones

U niversity College London

University of London

A thesis su b m itted for th e degree of

D o c to r o f P h ilo so p h y

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ProQuest Number: U644143

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To my Parents

for giving me the opportunity to learn and to understand.

To my Teachers

for showing me how to w rite it down.

To my Sister

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Acknowledgements

I would like to th an k Professor Frank Sm ith for his encouragem ent and guidance as

my supervisor, Dr. Sean O ughton and Dr. Sergei T im oshin for all th eir advice and

assistance, and th e Engineering and Physical Sciences Research Council for their

financial support. I would also like to th a n k all th e lads from th e UCL football

team for providing an escape from th e stu d y of ‘how aeroplanes fly’as they liked to

call it. Finally, I would like to th a n k my family and Loraine for th eir support and

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Abstract

An aircraft in low-level flight experiences a large increase in lift and a m arked re

duction in drag, com pared w ith flight a t altitude. This phenom enon is term ed the

‘w ing-in-ground’ effect. In these circum stances a region of high pressure is created

beneath th e aerofoil, and a pressure difference is set up betw een its upper and lower

surfaces. A pressure difference is not p e rm itte d a t th e trailing edge and therefore a

mechanism m ust exist which allows th e pressures above and below to adjust them

selves to produce a continuous pressure field in th e wake. It is th e stu d y of this

mechanism and its role in th e aerodynam ics of low-level flight th a t forms th e basis

of our investigation. We begin in C h ap ter 2 by considering th e flow p ast a th in aero foil moving a t m oderate distances from th e ground, th e typical ground clearance a

being of order unity. T he aforem entioned m echanism is introduced and described in

detail in th e context of this inviscid problem . C h ap ter 3 considers th e sam e flow for

large and small ground clearances and in th e later case shows th a t th e flow solution

beneath th e aerofoil takes on a p articu larly simple form. In th is case th e lift is shown

to increase as a~^. In C hapter 4 we focus on th e flow p ast th e trailing edge of an

aerofoil moving even nearer th e ground, w ith th e ground ju st outside th e boundary

layer. We show th a t in this case our asym ptotic theory for small a is consistent

w ith a ‘triple-deck’ approach to th e problem which incorporates ground effects via

a new pressure-displacem ent law. T he triple-deck ground-interference problem is

stated and solved. In C hapter 5 we investigate th e case where th e aerofoil is so near

th e ground th a t th e ground is inside th e boundary layer. Here th e moving ground

interacts w ith th e aerofoil in a fully viscous way and th e non-linear boundary layer

equations hold along th e entire length of th e aerofoil. Again a pressure difference at

th e trailing edge is not p erm itted and th is produces up stream adjustm ent back to

th e leading edge. Regions of reversed flow can occur and th eir effects, w ith regard to

downforce production and racing car u n d ertray design, are considered. In C hapter

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C ontents

1 In trod u ction 15

2 Inviscid ‘W in g-In -G rou n d ’ E ffect 27

2.1 In tr o d u c tio n ... 27

2.2 Problem F o r m u la tio n ... 29

2.3 M athem atical M e th o d s ... 31

2.3.1 Sym m etry A rgum ents ... 31

2.3.2 T he Solution ...32

2.3.3 Com pleting th e B oundary C o n d itio n s ...34

2.3.4 T he Pressure Difference Solution ... 42

2.4 N um erical M e th o d s ...43

2.4.1 T he E valuation of th e Pressure Difference [p] ( a : ) ... 43

2.4.2 T he Evaluation of -0 (x) 43

2.4.3 T he Evaluation of h { x ) and M { x , ^ )... 45

2.4.4 T he Evaluation of f (x) 46 2.4.5 The Evaluation of (p) (x) and {v) { x ) ...48

2.5 Flow P r o p e r tie s ...52

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C O N T E N T S 5

3 Inviscid S olu tion s for ^ 1 and ^ <K 1 57

3.1 I n tr o d u c tio n ... 57

3.2 Problem F o r m u la tio n ...60

3.3.1 Fourier Transform ing th e Integral E q u a t i o n s ... 60

3.4 Flying High > 1 ) ...62

3.4.1 T he Leading O rder Solution for ^ 1 ...63

3.4.2 T he Next O rder C orrection for Z$> 1 ...65

3.5 Flying Low (/5 1) ...66

3.5.1 T he Leading O rder Solution for <K 1 ...68

3.5.2 T he In-Between O rder Solution for /? <C 1 ...70

3.5.3 T he Next O rder C orrection for ^ 1 ... 71

3.6 Edge F l o w s ... 72

3.6.1 T he Trailing Edge N o n -E ig e n so lu tio n ... 73

3.6.2 The Leading Edge E ig en so lu tio n ... 83

3.6.3 Setting th e C onstants po- (0), P f - (0), and p i_ ( 0 ) ... 85

3.7 T he Global Solution for 1 ...88

3.8 Flow P r o p e r tie s ...89

4 V iscou s-In viscid ‘W in g -In -G ro u n d ’ Effect 91 4.1 I n tr o d u c tio n ... 91

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C O N T E N T S 6

4.3 M athem atical M e th o d s ...97

4.3.1 T he Solution of th e Interaction Law E q u a tio n s ...98

4.3.2 T he E valuation of th e Fam iliar Interaction L a w s ...100

4.3.3 T he Stream function-V orticity F o r m u la tio n ...102

4.4 N um erical M e th o d s ...104

4.4.1 The Basic D is c r e tis a tio n ... 105

4.4.2 The D iscretised W iener-Hopf Solution A l g o r i t h m ... 106

4.4.3 T he D iscretised Interaction L a w s ...109

4.4.4 T he Solution of th e B oundary Layer E quations ... 110

4.5 F urther C o m m e n t s ...116

4.6 Flow P r o p e r t ie s ... 117

5 V iscou s ‘W in g -In -G ro u n d ’ Effect 121 5.1 In tr o d u c tio n ... 121

5.2 Problem F o r m u la tio n ... 123

5.3.1 T he Flow D irectly B eneath th e Aerofoil ... 128

5.4 Num erical M e th o d s ...130

5.4.1 T he Basic D is c r e tis a tio n ... 131

5.4.2 T he Difference E q u a tio n s ... 132

5.4.3 T he Starting Profiles for 'ifj and Q, N ear x = 0 ... 133

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C O N T E N T S 7

5.4.5 Finding th e Pressure P- {x) ... 135

5.5 Flow P r o p e r t i e s ... 136

5.5.1 Wake Effects ...148

6 T unnel E ffects 155 7 C onclusions and Further W ork 161 7.1 Sum m ary ...161

7.2 F urther W o r k ... 163

A T h in Layers and th e Triple D eck S tru ctu re 165 A .l B oundary L a y e r s ... 165

A. 1.1 T he Blasius B oundary L a y e r ... 167

A .1.2 T he G oldstein Near W a k e ... 168

A .2 The Triple D e c k ... 170

A .3 Diffusion L a y e r s ... 174

B W ien er-H o p f C alcu lation s 177 B .l I n tr o d u c tio n ... 177

B.2 C alculating i? (A)_^ and R { K ) _... 178

B.3 C alculating th e C onstant A ...179

B.4 C alculating th e C onstant // 181

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C O N T E N T S 8

C T riple-D eck S tu d y P rob lem s 187

C .l I n tr o d u c tio n ...187

C.2 Flow P ast a Bum p on a F lat P late in Surface E f f e c t ...189

C.2.1 Problem F o rm u la tio n ... 189

C .2 .2 Small Bum ps {h <^1) 191

C.2.3 Larger B u m p s ...193

C.3 Flow P ast th e Trailing Edge of a F lat P la te in Sym m etric Tunnel Effect 196

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List o f Figures

1.1 T he KM Ekranoplan, designed by R otislav Alexiev, w ith its relatively short stubby wings and Y -shaped ta il...18

1.2 The KM E kranoplan as viewed from the side... 18

1.3 The X-112, designed by Alexander Lippisch, w ith its reverse

delta-wing planform ...19

1.4 The X-112 as viewed from th e side...19

1.5 The A m phistar designed by D im itri Sinitsyn. The m odern incarna

tion of th e KM ekranoplan 21

1.6 T he F lairb o at, desined by G unther Jorg, w ith its tan d em aerofoil

configuration...21

1.7 T he Airfisch 3 designed by Hanno Fischer. T he m ost m odern incar

nation of th e Lippisch X-112...22

1.8 T he Hover wing, also designed by H anno Fischer, w ith its unique cata

m aran /hovercraft take-off system ...22

2.1 Cross-section of an aerofoil of thickness e w ith two viscous boundary

layers and a th in viscous wake flying a t a ground clearance a ...33

2.2 Sym m etry argum ents lead to th e introduction of a v irtu al image aero foil beneath th e g r o u n d ... 33

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L I S T OF F IG U R E S 10

2.3 T he integration contours T+, T=, and T_ are defined in th e complex

C plane as above ... 35

2.4 T he pole in the integrand m ust be circum navigated when z is on the surface of th e a e ro fo il...35

2.5 T he functions / (x), h (a;), M (rr, {), a n d '0 (a:) for a = 0.5... 50

2.6 T he sums and differences [p] (a:), (p) (x), [u] (x), and (v) (x) for a = 0.5. 50 2.7 T he pressures and velocities p± (x) and v± (x) for a = 0.5... 50

2.8 The pressure p ertu rb atio n p { x , y ) for a = 0.5... 51

2.9 The transverse velocity p e rtu rb atio n v (x, p) for a — 0.5...51

2.10 The stream w ise velocity p e rtu rb atio n u (x, y) for a; = 0.5...51

2.11 The to ta l displacem ent thicknesses (x) of a horizontal fiat plate aerofoil in ground effect for a = 16 to T ...53

2.12 The pressures p± (x) on a horizontal fiat p late aerofoil in ground effect for a = 16 to T ...53

2.13 T he displacem ents (x) of a fiat p late aerofoil in ground effect at positive angle of a tta ck for o; = 16 to ^ ... 55

2.14 T he pressures p± (x) on a fiat p late aerofoil in ground effect a t positive angle of a tta ck for a = 16 to ^ ... 55

2.15 T he displacem ents ô± (x) of a fiat plate aerofoil in ground effect at negative angle of a tta ck for a = 16 to ^ ... 56

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3.2 T he ‘around-the-edge’ eigensolution which is p erm itted only a t the

leading edge (II) and th e stream lined non-eigensolution which may

occur a t either th e trailing edge (III) or th e leading edge...59

3.3 N um erical solutions for th e pressures (a;) and p_ (x) for a = ^ as

com puted using th e m ethod developed in C h ap ter 2... 90

3.4 A nalytic global solutions for the pressures (x) and p_ (x) ioi a = ^

as calculated using th e small (3 theory developed in this chapter. The

leading edge eigensolution has been added to th e global solution of

section 3.7, however th e non-eigensolutions have n o t... 90

4.1 T he triple deck flow stru ctu re a t th e trailing edge of an aerofoil in

ground effect. Shows th e lower decks (I±), th e m ain decks (II±), th e

upper deck (III), and th e ground diffusion layer (IV) ... 93

4.2 T he interactive ground effect mechanism. Pressure continuity, in the

wake, drives a displacem ent effect which in tu rn drives a pressure

correction and so o n ...93

4.3 T he triple deck pressures P± (V ) for a flat p late in ground effect for

Q = 16 to ^ ... 118

4.4 T he triple deck displacem ent gradients A'^ (X ) for a flat plate in

ground effect for â = 16 to ^ ... 118

4.5 T he triple deck displacem ents A± (X ) for a flat p late in ground effect

for â = 16 to ...119

4.6 T he norm alised skin frictions (X, 0) on th e trailin g edge of a flat

p late in ground effect for â = 16 to ^ ...119

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L I S T O F F IG U R E S 12

5.2 Cross-section of a th in aerofoil in extrem e ground effect. T he bound

ary layers associated w ith th e aerofoil and th e ground merge produc

ing a fully viscous solution beneath th e aerofoil... 124

5.3 The linearly expanding diffuser shape w ith front wheel (not to scale). 137

5.4 The proposed step change in height diffuser shape, which according

to lubrication theory produces more downforce th a n th e linearly ex

panding one... 137

5.5 Several examples of stream wise velocity profiles u ( x , y ) , a t varying

X stations, for forward flow at d = 1, ^ = 1/2, and 7 = 1 /3 ... 138

5.6 Several examples of stream wise velocity profiles u { x , Y ) , a t varying

X stations, for a reversed flow case a t d = 1, /? = 2, and 7 = 1 / 3 .. . . 138 5.7 The stream lines of constant ' ip{x,Y) for d = 16 to /3 = 1, and

7 = 1 directly beneath th e diffuser ...140 5.8 The pressures p± (x) for d = 16 to ^ = 1, and 7 = ^ directly

b eneath the diffuser. Note th e changes of scale betw een each row of

g raphs... 141

5.9 The skin frictions {x) and flo (3;) for d = 16 to ^ , ^ = 1, and

7 = 1 directly beneath th e diffuser ...141 5.10 The stream lines of constant 'ip{x^Y) for d = 1, /3 = 16 to and

7 = 1 directly beneath th e diffuser ... 143 5.11 The pressures p± (æ) for d = 1, ^ = 16 to and 7 = ^ directly

b eneath th e d iffu se r... 144

5.12 The skin frictions [x) and LIq(3:) for d = 1, ^ = 16 to and

7 = 1 directly beneath th e diffuser ... 144 5.13 The stream lines of constant i p ( x ^ Y) for d = 1, P = 1, and 7 = 0 to

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5.14 T he pressures (x) for a = 1, /? = 1, and 7 = 0 to 1 directly beneath th e diffuser ... 147

5.15 T he skin frictions (x) and Ho (a;) for S = 1, jS = 1, and 7 = 0 to 1 directly b en eath th e d if f u s e r ... 147

5.16 The non-dim ensional lift L as a function of th ro a t length 7 and expan sion param eter ^ as calculated as p a rt of m any num erical boundary-

layer s o lu tio n s ... 149

5.17 T he non-dim ensional lift L as a function of th ro a t length 7 and ex pansion param eter /3 as calculated using lubrication t h e o r y ... 149

5.18 Wake results showing stream lines, pressures, and skin frictions for

forward and separated flows. Notice th e reduced pressure variation

in th e right hand case... 151

5.19 Stream lines of constant Tp {x^Y) showing th e extrem ely small closed

eddy close to th e trailing edge... 152

5.20 T he flow topology close to th e trailing edge as observed in the nu

m erical wake solutions (not to scale)...154

5.21 A sim ilar flow topology envisaged for vortex shedding behind a cir

cular cylinder... 154

6.1 Cross-section of a th in aerofoil w ith b oundary layers and a th in vis cous wake moving through a tu n n e l... 158

6.2 T he doubly periodic cascade of image aerofoils introduced due to

sym m etry conditions being applied on the tu n n el floor and ceiling. . . 159

6.3 Integration contours T2n± in th e complex plane corresponding to the

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A .l T he solution to the equation / " ' (77) + ( 77) / " (77) = 0 w ith / (0) = 0, / ' (0) = 0, and / ' ( +00) = 1... 169

A.2 The solution to th e equation g”' (77) + | p (77) (77) — (77)^ = 0 w ith

g (0) = 0, g" (0) = 0, and g" ( +00) = A = 0.3346... 169

C .l The triple-deck flow stru ctu re around a bum p on an otherw ise flat

surface, which is moving past another surface... 188

C.2 The triple-deck flow stru ctu re near th e trailing edge of a th in aerofoil in sym m etric tunnel effect... 188

C.3 T he pressure d istribution P [ X) on a small bum p in surface effect,

for a range of surface clearances â. h = 0.01. G ( X ) = 194

C.4 The displacem ent A [ X ) on a small bum p in surface effect, for a range

of surface clearances â. h = 0.01. G { X) = 194

C.5 The pressure distribution P { X) on a larger bum p in surface effect,

for a range of surface clearances â. h = 1. G { X) = (1 — X' ^Ÿ for |A | < 1 and 0 otherw ise...195

C.6 The displacem ent A (A ) on a larger bum p in surface effect, for a range of surface clearances â . h = 1. G (A ) = (1 — A^)^ for |A | < 1 and 0

otherw ise... 195

C.7 The pressure P (A ) over th e trailing edge of a flat p late in sym m etric

tunnel effect, for a range of surface clearances â... 197

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C hapter 1

In trod u ction

Is it a boat? Is it a plane? Is it a car? No, i t ’s a W IG. This is th e term given to wing-

in-ground effect vehicles which are designed to exploit th e aerodynam ic efficiency

gains associated w ith low level flight.

Ever since th e beginning of m anned flight, pilots have experienced som ething strange

when landing aircraft. Ju st before touchdown it feels as if th e aircraft doesn’t want

to land, as if it is floating on a cushion of air; this phenom enon is term ed ‘wing-in-

ground effect’ or simply, ‘ground effect’.

A erodynam ically speaking, two things happen as an aircraft approaches the ground

and these two phenom ena are term ed span-dom inated and chord-dom inated ground

effect. T he former results in a reduction in th e induced drag, D , and th e la tte r

results in an increase in th e lift, L, experienced by th e aircraft.

The two m ain sources of drag experienced by aircraft in flight are referred to as

th e skin drag and th e induced drag. As th e nam e suggests th e first is caused by

friction of th e air on the skin of the aircraft, w hilst th e second is produced as a

direct result of th e w ing’s ability to generate lift and is som etim es called the ‘lift

induced d ra g ’. W hen a wing generates lift th e high pressure air, created beneath the

wing, leaks around th e wing tip to m eet th e low pressure air on to p of th e wing and

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C H A P T E R 1. IN T R O D U C T I O N 16

causes a wing-tip vortex. These vortices are som etim es visible w hen w ater in the

air condenses in th e low pressure vortex core and ap p ear as spiral lines extending

backw ards from th e wing tips. T he energy th a t is stored in these vortices is lost and

is experienced by th e aircraft as drag. Hence th e te rm ‘lift induced d ra g ’.

W hen th e aircraft approaches the ground these w ing-tip vortices become weaker

due to destructive interference between th e vortices them selves and th eir counter-

ro ta tin g image vortices beneath th e ground. This leads to a corresponding reduction

in th e induced drag on th e aircraft.

As m entioned earlier chord dom inated ground effect increases lift. W hen in ground

effect, th e air passing beneath th e wing is slowed down so th a t th e pressure there

rises causing a large increase in lift; this is sometim es referred to as ‘ram effect’. In

some circum stances th e fluid beneath th e aerofoil can be m ade to stag n ate producing

large pressures, leading to large lift.

T he combined result of th e two phenom ena described above is to increase th e ratio

L /D , which is commonly used to m easure th e efficiency of aircraft since when in

stead y flight weight is equal to lift and th ru s t is equal to drag and so L / D is an

expression of how much weight can be carried for a given am ount of th ru st. In

fact, lift-to-drag ratios for wings in ground effect are roughly twice as good as those

for wings in free flight, making wing-in-ground effect flight one of th e m ost energy

efficient m ethods of tra n sp o rta tio n available.

This thesis is concerned w ith identifying and und erstan d ing th e underlying aerody

nam ic mechanisms which give rise to chord-dom inated ground effects for th e entire

range of possible ground clearances from flight a t a ltitu d e to surface skimming.

T h e phenom enon of wing-in-ground effect has been known since 1920, in fact the

W right B rothers unknowingly used ground effects in th e ir first atte m p ts to fly in the

1900’s and th e first theoretical investigation of ground effect was m ade by Wiesels-

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well aware th a t if one of their engines was dam aged, m aking sustained flight a t al

titu d e impossible, an alternative to ditching in th e ocean was to descend alm ost to

sea level and use wing-in-ground effect to get hom e on th eir one rem aining engine.

Large birds, such as th e A lbatross, are also known to utilise wing-in-ground effect

to conserve energy on long flights.

In 1935 the Finnish engineer K aario (1959a,b) was th e first to build a vehicle specifi

cally designed to take advantage of ground effects. However it w asn’t u n til the 1960’s

th a t independent research in m any countries including th e USSR, USA, Japan, and

G erm any began to take off.

In th e USSR th e m ajor developm ents took place a t th e C entral Hydrofoil Design

Bureau, led by R otislav Alexiev. T he need for faster tra n sp o rta tio n over w ater led

Alexiev to consider wing-in-ground effect vehicles as an im provem ent upon his earlier

invention, th e hydrofoil. His work eventually led to th e developm ent of th e 540

tonne KM E kranoplan dubbed th e ‘C aspian Sea M onster’ by A m erican intelligence

officers after they sp otted its peculiar planform -shape in satellite images. The KM is

characterised by its relatively short stubby wings and its huge Y -shaped tail, which

operates out of ground effect and gives th e craft stability. It was approxim ately

100 m etres long and travelled at an altitu d e of 20 m etres above th e C aspian Sea at

around 300 mph! See Figures 1.1 and 1.2.

A round th e sam e tim e th e G erm an aerodynam iscist Lippisch (1964), inventor of

th e delta-w ing, designed a revolutionary new W IG; th e X -11 2. It h ad a reversed delta-w ing planform w ith negative dihedral a t th e leading edge and a large T-shaped

tail, again for stability. This configuration proved to be inherently stable in ground

effect and m any recent designs have been based on th e original Lippisch concept.

See Figures 1.3 and 1.4

More recently, fu rth er developm ent in th e USA, Jap an , Germany, and A ustralia

have lead to m any weird and wonderful W IG designs. For exam ple, th e A m phistar

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C H A P T E R 1. IN T R O D U C T IO N 18

a

Figure 1.1: The KM Ekranoplan, designed by Rotislav Alexiev, w ith its relatively

short stubby wings and Y-shaped tail.

tM M m

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C H A P T E R L IN T R O D U C T IO N 19

Figure 1.3: The X-112, designed by Alexander Lippisch, w ith its reverse delta-wing

planform.

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worked w ith Alexiev on th e C aspian Sea M onster project; th e F lairb o at designed by

G unther Jorg (Jorg (1987, 1997)) w ith its tan d em wing configuration; th e Airfisch 3

designed by H anno Fischer (Fischer (1989, 1988)), th e m ost m odern re-incarnation

of the Lippisch; and th e Hoverwing also designed by H anno Fischer (Fischer and

M atjasic (1996, 1997)), which combines catam aran and hovercraft technology to

aid take off; this being im p o rtan t since initially gettin g free of th e w ater, and into

ground effect, efficiently is one of the m ajor practical difficulties associated w ith

wing-in-ground effect flight. See Figures 1.5, 1.6, 1.7, and 1.8.

It is our aim throughout this thesis to gain an understanding of th e physical mech

anisms involved in producing aerodynam ic ground effects, these being th e fluid dy

nam ical phenom ena th a t are introduced when one forces an aerofoil to perform

in close proxim ity to th e ground. In fact we will only consider th e lift enhancing

properties of wing-in-ground effect flight.

As m entioned earlier, in ground effect a region of high pressure is created beneath

the aerofoil and a pressure difference is set up betw een its upper and lower surfaces;

this pressure-difference is responsible for th e increased lift experienced by aircraft

when travelling near th e ground. For th in tw o-dim ensional aerofoils th e lift, L, per

unit length is given by

1

L =

J

{ p _ { x ) - p + { x ) ) d x - \ - - - , (1.1)

0

to leading order, where the leading and trailing edges of th e aerofoil are a t x = 0

and 1 for convenience and th e functions (æ) and p_ (æ) denote th e pressures on the upper and lower surfaces of th e aerofoil respectively. As one can clearly see it is

th e difference in th e pressures (æ) and (x) th a t creates th e lift.

However, a difference in pressure is not p erm itted at th e trailing edge, due to th e fact

th a t th e pressure may not vary across th e aerofoil’s th in viscous wake, and therefore

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Figure 1.5: The A m phistar designed by D im itri Sinitsyn. The modern incarnation

of the KM ekranoplan.

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Figure 1.7; The Airfisch 3 designed by Hanno Fischer. The most modern incarnation

of the Lippisch X-112.

Figure 1.8: The Hoverwing, also designed by Hanno Fischer, w ith its unique cata

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to ‘feel’each other and adjust themselves accordingly so as to produce a continuous

pressure field a t th e trailing edge and in th e wake.

It is the study of this mechanism and its role in th e aerodynam ics of th in aerofoils

travelling parallel w ith and close to th e ground th a t forms th e basis of th e following

investigation.

We begin, in C hap ter 2, by considering th e inviscid p o tential fiow p ast a th in aerofoil flying a t relatively large distances from th e ground. T he aforem entioned mechanism

is introduced and described in th e context of this classical th in aerofoil problem

(Sedov (1965)). A great deal of work has been focused on this and sim ilar problems

in two and th ree dimensions and a t large and sm all ground clearances including

significant contributions by W idnall and Barrows (1970), Tuck (1971), K ida and

Miyai (1973), Tuck (1980), P lotkin and Kennell (1981), Newm an (1982), Tan and

Plotkin (1986), and P lotkin and Dodbele (1988).

However, in contrast to all of th e above work we include th e effects of viscosity by

considering th e infiuence of th e attach ed Blasius boundary layers (Blasius (1908))

and th e aerofoil’s th in G oldstein wake (G oldstein (1930)) on th e inviscid outer fiow.

This is achieved by effectively incorporating th e displacem ent thicknesses of th e th in

viscous layers into th e function which describes th e aerofoil shape. T he inclusion

of these effects is essential since th e singular n a tu re of th e Blasius and Goldstein

displacem ent thicknesses greatly infiuence th e lift on th e aerofoil in ground-effect,

especially when sm aller values of th e ground clearance p aram eter a are considered.

Having included these viscous effects th e boundary condition on th e ground is re

placed by a sym m etry condition, which leads to th e in tro d u ctio n of an image aerofoil

beneath th e ground. The problem of finding th e fiow solution is th en reduced, using

complex analytical techniques (G arrier et al. (1966)) to th a t of solving a singular

integral equation for th e lift distribution on th e aerofoil. T he relevant solution is

obtained analytically (Kondo (1991)) and is evaluated num erically for a range of

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In C hapter 3 we examine th e large and small ground clearance lim its of th e ground-

interference problem introduced in C hapter 2. We begin w ith th e integral equations derived in C h ap ter 2 and use Fourier integral transform techniques (Sneddon (1972))

to simplify th e integral equations in each limit.

For large ground clearances we are able to show th a t th e aerofoil does not feel the

presence of th e ground to leading order and a t next order th e effect of th e ground’s

presence is to add a v irtu al angle of a tta ck to th e aerofoil.

For small ground clearances we show th a t th e flow solution directly beneath the

aerofoil takes a very simple form, which depends ultim ately on th e flow solution

in two scaled regions around th e leading and trailing edges. Two classes of edge-

flow solutions are identified and application of the K u tta condition a t th e trailing

edge allows a unique solution to be constructed in a logical way. T he W iener-Hopf

technique (Noble (1958), Sparenberg (1956), and Sparenberg (1958)) is used to find

one class of edge-flow solution w hilst a conformai m apping technique (Churchill

and W ard Brown (1990)) is employed to find th e other. T he lift on th e aerofoil is

crucially shown to increase as ea~^ for small ground clearances a , where e =

In C hapter 4 we focus in on the trailing edge region and consider this edge-flow

when th e aerofoil is even nearer th e ground, w ith th e ground ju st outside th e clas

sical boundary layer. In this case th e inviscid asym ptotic th eo ry for small ground

clearances, developed in C hapter 3, is entirely consistent w ith a ‘triple deck’ flow

stru ctu re at th e trailing edge (Stew artson (1969) and M essiter (1970)), which in

corporates ground effects via a new pressure-displacem ent interaction law. This

consistency is entirely due to th e fact th a t viscous effects were included in the orig

inal inviscid problem and it is now clear why these effects m ust be included from

th e outset if one is to construct a sensible theory.

T he triple deck ground-interference problem is s ta te d and solved in p a rt analytically

and in p a rt num erically using a W iener-Hopf solution of th e new pressure displace

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C H A P T E R 1. I N T R O D U C T I O N 25

C arter (1979). Solutions for th e pressures and displacem ents are presented for a

range of ground-clearances and shown to agree, for large ground-clearances, w ith

th e results of Jobe and B urggraf (1974) for flow p ast th e trailin g edge of a flat p late

in th e absence of th e ground. The new pressure-displacem ent law for flows w ith

ground interference takes on a particularly simple form for sm all ground clearances

and th e im plications in term s of th e solutions presented are discussed.

In ch ap ter 5, we consider th e flow past a th in aerofoil moving extrem ely close to

th e ground, w ith th e ground inside th e classical b oundary layer. T h e governing

equations inside and outside the gap are shown to be th e boundary layer equations

and th e moving ground can now strongly influence th e flow solution in a fully vis

cous way. In contrast w ith the work of Tuck and Bentwich (1983), Tichy and Chen

(1985), Tichy (1986), Szeri (1987), and W ilson and Duffy (1998) we concentrate

on th e flow in a diverging channel where th e m inim um ground clearance is tow ards

th e front of th e aerofoil. T he diverging channel case typically produces negative

lift or downforce and applications in Formula One racing car design are considered.

T he mechanisms encountered in th e context of these flows include a viscous-inviscid

in teraction which fllls th e entire gap and is again associated w ith th e requirem ent

of pressure continuity a t the trailing edge, th e generation of strong upstream influ

ence which forces a localised pressure jum p a t th e leading edge, and im portantly

su b stan tial flow separation and reversed flow a t th e trailing edge leading to certain

wake effects. T he governing equations are solved num erically and th e solutions are

shown to agree w ith th e predictions of the inviscid asym ptotic th eo ry of C hapter 3

for larger gaps and those of lubrication theory (Batchelor (1967)) for sm aller ones.

Finally, in C h ap ter 6 we consider th e effects of adding a ceiling into th e problems discussed in C hapters 2, 3, and 4, thus effectively considering w ing-in-tunnel effects. We show th a t th e results of C hapters 2 ,3 , and 4 can be generalised to include tunnel

effects by simply replacing th e algebraic kernels appearing in th e integral equations

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moving through tunnels can th en be obtained in m uch th e sam e way as for those of

th in aerofoils in ground effect.

Therefore, in th e investigation th a t follows we present a com plete description of

th e phenom enon of two-dim ensional chord-dom inated “w ing-in-ground”effect, as ap

plied to th in aerofoils, spanning th e entire range of ground clearances from those of

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C hapter 2

Inviscid ‘W in g-In -G rou n d ’ Effect

2.1 In tr o d u c tio n

As m entioned already an aircraft in low-level flight can experience a large increase

in lift and a m arked reduction in drag, com pared w ith flight a t altitude. This phe

nomenon is term ed ‘w ing-in-ground’ effect. In this chapter we consider th e increase

in lift produced by essentially inviscid fluid mechanics, on a th in aerofoil travelling

near th e ground . The inviscid assum ption is valid for th e case in which th e ground

does not directly interfere w ith th e viscous boundary layers produced on th e aerofoil

surfaces.

One can consider this chapter as a prelim inary stu d y of ‘w ing-in-ground’ effect,

specifically a stu dy of th e phenom enon in its weakest form. T h e inclusion of this

chapter allows us th e o p p ortunity to introduce some of th e im p o rtan t physical ideas

and m ath em atical techniques, which will be developed in th e later chapters, in the

familiar context of a potential flow problem. In fact th e ‘ground w ork’ done in this

chapter allows us to approach th e problem s of th e later chapters more readily, in

the knowledge th a t th e assum ptions made later are based on solid foundations.

Inviscid p o ten tial flow theory predicts th e occurrence of a rapid variation in pressure

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C H A P T E R 2. IN V ISC ID ‘W IN G -IN -G R O U N D ’ E F F E C T 28

near the trailing edge of certain slender aerodynam ic shapes, when placed in a

uniform stream . It is th e precise n atu re of this pressure variation which gives rise to

th e ‘Triple-Deck’ theory of trailing edge flow due to Stew artson (1969) and M essiter

(1970). If we are to extend triple-deck theory to include th e effects of ground-

interference we m ust first un d erstan d th e aforem entioned pressure variation in the

context of a th in aerofoil moving near th e ground.

Therefore we begin our investigation by considering th e flow p ast a th in aero

foil travelling at m oderate distances from, and parallel to, th e ground. We non-

dimensionalise the lengths and velocities in th e problem on L and U respectively

where L is th e aerofoil chord length and U is th e aerofoil’s speed relative to the

ground. As a result the fluid pressure is non-dim ensionalised based on the quantity

pU"^ where p is th e fluid density. We will lim it th e discussion by considering only

two-dimensional, incompressible, steady flows a t high Reynolds num ber and will

consequently take as our startin g point th e 2D steady Navier-Stokes equations in

non-dim ensional form as w ritten below.

+ + (2.1)

+ + + (2.2)

£ + ^

(2

3 )

The stream w ise and transverse com ponents of th e velocity vector field are denoted

by U {x, y) and V (x, y) respectively and th e scalar pressure field is denoted P (x, y).

The non-dim ensional Reynolds num ber is defined to be R e = where u is the

kinem atic viscosity of th e fluid. We will assume Re ^ 1.

In a frame moving w ith the aerofoil, our task becomes one of determ ining the flow

around a th in body in a uniform stream , close to a plane boundary which is moving

at the free stream velocity. See Figure 2.1.

We denote th e typical non-dim ensional distance betw een th e ground and th e aerofoil

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C H A P T E R 2. IN V IS C ID ‘W IN G -IN -G R O U N D ’ E F F E G T 29

above inviscid ground-interference problem. However, we m ust include th e effects of

viscosity by incorporating boundary conditions into th e problem which are consistent

w ith th e presence of th in boundary layers adjacent to th e aerofoil, followed by a

th in viscous wake. The inclusion of th e wake, of unknow n shape g (a;), allows the

possibility of a discontinuous velocity field across th e wake centerline w hilst retaining

th e need for a continuous pressure field there. Also, by allowing this layer the

freedom to adjust its shape, we introduce th e m echanism by which the pressures

above and below the plate, p+ (a;) and p_ (x), m ay adju st them selves so as to meet

th e required pressure continuity condition at th e trailing edge, (1) = p_ (1).

Finally we solve th e governing po ten tial fiow equations, subject to th e appropriate

boundary conditions, using complex analytical techniques for a range of ground

clearances. At th e h eart of th e problem lies a singular integral equation which

m ust be solved for th e lift distrib u tio n on th e aerofoil. Once this all-im portant

lift distrib u tio n has been found th e fiow solution can be com pleted. T he lift is

qualitatively shown to increase as for decreasing ground clearance a and the

rapid variation in pressure predicted a t th e trailing edge for flows w ithout ground-

interference is shown to persist in flows close to th e ground.

2.2 P r o b le m F o rm u la tio n

We begin by studying th e flow p a st a general th in aerofoil, w ith b o th cam ber and

thickness, moving parallel w ith and close to th e ground. Since we are dealing w ith

a th in aerofoil we expect th e flow to differ only slightly from th a t of an undisturbed

uniform stream , alm ost everywhere, and therefore construct th e solution in term s

of th e following p ertu rb atio n expansions.

U { x , y ) = l - \ - £ u ( x , y ) - \ , (2.4)

V { x , y ) =-- 0- \ - e v { x , y ) - \--- , (2.5)

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T he expansion param eter is the classical boundary layer thickness and typical aero

foil thickness, e = . The expansions above are expected to hold everywhere

except in th e boundary layers, th e wake, th e ground layer and near th e leading and

trailin g edges. The flows in these regions are considered in A ppendix A. The bound

ary layers and wake are of Blasius and G oldstein typ e respectively, th e ground layer

is a passive linear diffusion layer and th e flow in th e neighbourhood of th e trailing

edge is described by th e triple-deck theory of Stew artson (1969) and M essiter (1970).

T he leading edge flow is not considered.

We su b stitu te expansions (2.4)-(2.6) into th e Navier-Stokes equations and obtain,

a t leading order, th e inviscid equations of linearised p oten tial flow.

(2,8)

(^> 3^) + ^ (^> 2^) =

(2.9)

A fter th e elim ination of u (x, y) these become th e C auchy-R iem ann equations in the

unknow n pertu rb atio n s p (x, y) and v (x, y),

2 (^.2/) = (2-10)

(2.11)

O ur concern, then, is w ith finding th e complex function w { x + iy) = p( x ^ y ) -f

iv (T, y), analytic in a slit upper half plane, bounded in th e far field, and satisfying

th e boundary conditions,

w [ x P 0%) = p+ (z) + iv+ (a;), (2.12)

w { x — Iii) = p - { x ) i v - { x ) , (2.13)

w [x — iq) = p={x)-\ -Qi , (2.14)

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C H A P T E R 2. IN V IS C ID ‘W IN G -IN -G R O U N D ’ E F F E C T 31

where (x) is ground pressure. D isplacem ent effects produced by th e aerofoil

profile, th e Blasius boundary layers on th e aerofoil surfaces, and th e G oldstein wake

are included via th e condition on (x) and u_ {x) given below (see A ppendix A ).

s' {x) for a: G (—0 0,0)

(a;) = < c' (x) ± (x) ± % (x) for X G [0,1] (2.16)

s '( x ) ± (x) for x G ( 1 , +oo)

where the Blasius and G oldstein displacem ent thicknesses are denoted sSb (x) and

sôg (x) respectively, the cam ber and thickness of th e aerofoil are denoted ec (x) and

et (x) respectively, and th e unknow n shape of th e dividing stream line upstream

and th e defiected wake dow nstream are denoted e s ( x ) . We also define th e to ta l

displacem ent thicknesses eô± (x) by w riting

X

s±{x)

= (2.17)

0

2.3 M a th e m a tic a l M e th o d s

2 .3 .1

S y m m e tr y A r g u m e n ts

We simplify th e problem slightly by elim inating p = (x ). T his is achieved by re

interpreting th e no-penetration condition im posed a t th e plane boundary, z = x —ia,

as a sym m etry condition. We are th e n faced w ith th e problem of finding th e complex

function w {x iy) = p (x, y) + iv (x, y) which is analytic in a doubly slit plane,

bounded in th e far field, and satisfies th e new boundary conditions,

w (x + Oi) = (x) -h i v ^ ( x ) , (2.18)

w { x — Oi) = (x )-h ( x ) , (2.19)

w { x — i(3 + Oz) = p - (x) — IV- ( x ) , (2.20)

w { x — i(5 — Oz) = (x) — ZU+ ( x ) , (2.21)

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where 15 = 2a. Due to these sym m etry argum ents we have introduced an image

aerofoil beneath th e ground and are now essentially considering th e flow p ast an

unstaggered, non-lifting biplane w ith two th in viscous wakes. See Figure 2.2.

2 .3 .2

T h e S o lu tio n

We flnd a solution by applying C auchy’s integral form ula for w (z) three times, using

th e contours F+, F=, and F_ in th e complex ( plane deflned in Figure 2.3.

Taking th e lim it as lim , summing th e resulting balances, and im posing th e

bound-R —>oo

ary conditions (2.18)-(2.21) yields the solution

(( — z)

where for convenience we have introduced th e n o tatio n

[u] (a;) = v+ (x) - ( x ) , (2.24)

{v) (a:) — u+ (a:) 4- u_ (a;), (2.25)

[p] (3:) = P+ (x) - p - (a;), (2.26)

(p){x) = p + { x ) T p - { x ) , (2.27)

for sums and differences. We can flnd th e pressure an d transverse velocity p e rtu r

bations, p { x , y ) and v (a;,?/), by taking real and im aginary p a rts of equation (2.23)

to obtain

^

=

è l

—oo

(

+00

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60 (x)

■{>

4>

Figure 2.1: Cross-section of an aerofoil of thickness £ w ith two viscous boundary

layers and a th in viscous wake flying a t a ground clearance a

r>

Figure 2.2: Sym m etry argum ents lead to th e intro d u ctio n of a v irtu al image aerofoil

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- i l

(2.29)

T he stream w ise velocity p e rtu rb atio n u {x, y) is th en found via th e simple relation

u { x , y ) = - p { x , y ). (2.30)

The solution is therefore determ ined in term s of th e differences [u] {x) and [p] (æ).

However, although we know th e function [u] {x) everywhere and th e pressure con

tinuity condition (2.22) informs us th a t [p] (x) = 0 outside th e interval x G [0,1],

[p] (x) is unknow n inside this interval. T he function [p] ( z ) in th e interval x G [0,1]

describes th e lift distribution on th e aerofoil and is therefore of prim ary im portance.

It m ust be found before th e solution is complete.

2 .3 .3

C o m p le tin g t h e B o u n d a r y C o n d itio n s

We now tu rn to th e task of com pleting th e boundary conditions by finding the

pressure difference [p] (x) in th e interval x G [0,1]. Using C auchy’s integral formula

once again, we consider th e solution at a point on th e aerofoil surface. We redefine

th e contours T+ and T= slightly by adding a small circum navigation of th e pole

which is now encountered in th e integrand a t this point. See Figure 2.4 .

Having done this we continue as before by applying C auchy’s integral formula three

tim es using th e new contours F+ and F= for w {x + Oz) and w { x — Oi) respectively.

Taking th e lim it j i m lirn and sum m ing th e three resulting balances gives

u ,(x + 0 i ) + c . ( x - 0 i ) = +

m J (ç — X )

+ 00

(^ - t ) - i/3

(2.31)

1 r io — i/3 + Oi) — uj — i/3 — Oi)

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Im^ z = X + ly

+R

o

w(C) = p. + iy

P = 2a _ReC

o

w(C) = p. - iv

C = ^ - 2ia

w (0 = p+ - iy.

Figure 2.3: T he integration contours F+, F=, and F_ are defined in th e complex (

plane as above

Im^

z =

+R

I w(C) = p + iy

P = 2a _ReC

<y ■o

w(C) = p - iy

C = Ç - 2ia

w(C) = p+ - iy.

Figure 2.4: T h e pole in th e integrand m ust be circum navigated when z is on the

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Applying th e boundary conditions (2.18)-(2.21) and using th e sho rth an d notation

for sums and differences we arrive a t th e expression

TTl J

K - 4

(b] ( 0 - « M ( 0 ) (K - ^) + i/3) a

-(2.32)

If we take im aginary and real p arts of this expression we o b tain a pair of coupled

integral equations involving th e sums and differences, [v] (x), {v) (a;), [p] {x), and

(p) (x), namely

(v) (x) =

^ J

H ( 0 - TO(Ç - a;)j b] (Ç)

bX^;) =

^ f (^ j^ ^ ^ + m { ^ - x ) ‘j [ v ] { ^ ) d ^ - J l { ^ - x ) [ p ] { ^ ) d ^ ,

(2.33)

(2.34)

where I (x) — and m (x) = X

In th e above equations (5 = 2a where a is th e non-dim ensional ground-clearance

param eter. For convenience we will now adopt P as our p aram eter of choice. It

is im p o rtan t since it is th e distance between th e aerofoil and its v irtu al p artn er

beneath the ground. See Figure 2.2

The two integral equations m ust be solved subject to th e m ixed displacem ent and

pressure-continuity boundary conditions,

0 for æ E (—oo, 0)

t ' (x) 4- 2% (x) for X G [0,1]

2ôg{x) for x G ( 1 ,+ o o )

2c' {x) for X e [0,1]

? for a; 0 [0,1] b ] ( x ) = <

(v) (x) =

(2.35)

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[p] (s) =

(p) (z) =

? for T G [0,1]

0 for T 0 [0,1]

? for X G [0,1]

? for T ^ [0,1]

where ? is used to highlight th e fact th a t th e functions in question are unknown in

th e corresponding intervals shown above.

(2.37)

(2.38)

T h e S olu tion o f th e Integral E quations

It is clear from an exam ination of the integral equations and bou n d ary conditions

above th a t we m ust use our knowledge of {v) {x) in th e interval x G [0,1] to find

[p] (x) in th e interval x G [0,1]. We can accomplish this by considering integral

equation (2.33) for x G [0,1]. We apply th e pressure-continuity condition [p] (x) = 0

for X 0 [0,1] and, after a m inor rearrangem ent, equation (2.33) becomes th e integral

equation

( ( g - a )

[p

1 ^ J

(2.39)

valid for x G [0,1]. This is a singular Predholm integral equation of th e first kind

for the pressure difference, [p] (x), in th e interval x G [0,1]. T he right hand side

of th e equation is known. T he equation has a Cauchy type kernel and we refer to

Muskhelishvili (1946) for advice on its solution.

In order to solve th e equation we m ust first reduce it into an integral equation

of th e second kind which is easier to solve. This reduction involves finding the

solution of th e related ‘d o m in an t’ equation. T he dom inant integral equation adm its

more th a n one solution and we m ust choose th e one th a t is consistent w ith the

physically relevant constraint of zero pressure difference at th e trailing edge. The

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th e resulting Predholm equation of the second kind is solved for [p] (x) in th e interval

X G [0,1].

Once [p] (x) is determ ined we can explicitly determ ine {v) (x) and (p) (x) every

where using th e integral expressions (2.33) and (2.34) respectively. Furtherm ore

once th e sums and differences, [p] (x), (p) (x), [u] {x), and {v) (x) are all completely

determ ined, we m ay calculate p± {x) and v± {x) from th e simple relations

P±( ^) = ^ ((p) W ± [p] (2:)), (2.40)

% (3:) = ^ ((u) (x) ± M ( x ) ) . (2.41)

T h e Singular Integral E q u ation o f th e F irst K ind

We have reduced th e problem under consideration to th a t of finding the solution of

an integral equation subject to th e additional constraint th a t th e solution m ust be

zero at ac = 1. T h e integral equation (2.39) can be w ritten as

^ / ( ( 1 ^ ^ - ” » ( ? - j [P] (?) = f ( ^ ) . (2 42)

for T G [0,1] where

1

7

f { x ) = - / l { ^ - x ) [u] ( 0 - {v) ( x ) , (2.43)

7T J

—00

is a known function of x in th e interval x G [0,1].

The kernel of equation (2.42) is split into a singular p a rt and a regular p art. We

recognise th e singular p a rt as the C auchy-H ilbert kernel, and refer to it as the

dom inant p a rt of th e kernel. We rew rite (2.42) as

-

-f

=

f{x)

+ -

f

x)\p]

( 0 d^, (2.44)

7T J [ t — X ) 7T J

0 ^ ^ 0

for x G [0,1] by taking th e regular p a rt of th e kernel onto th e right-hand side. We

then proceed by seeking the solution of this dom inant equation as if th e right-hand

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T h e S o lu tio n o f th e D om in an t E q u ation

T he solution of th e dom inant equation

= (2.45)

0

is now considered. It is not uniquely defined by equation (2.45) and we m ust identify

th e ap p ro p riate solution by im posing th e constraint of zero pressure difference at

th e trailin g edge, [p] (1) = 0. T he solution we require is w ritten below for x G [0,1].

For th e details of its derivation see M uskhelishvili (1946, section 113).

We now use th e above solution to reduce th e original singular Fredholm integral

equation of th e first kind into a non-singular integral equation of th e second kind.

T h e R e d u c tio n o f th e Integral E q u ation

We apply th e integral operato r defined by equation (2.46) to b o th sides of the

dom inant equation (2.44) to obtain

1

X

h(x)-\~ - I M (x, g) [p] (x)

7T J (2.47)

for X G [0,1] where

'^ (^) = - ^ / /r?5' (fS)

and

M { x , 0 = (2-49)

In short, we have undone th e effects of th e C auchy-H ilbert operator on th e left of

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E quatio n (2.47) is nothing more th a n a Predholm integral equation of th e second

kind for th e pressure difference [p] (æ) in th e interval x G [0,1]. Before we consider

th e solution of (2.47) we make a minor alteratio n w ith regard to th e square root

singularities appearing in equations (2.47)-(2.49). If it is our intention to evaluate

the solution num erically th en these square root singularities could cause problems.

Therefore we factor th e square root out of equation (2.47) and thereby make sure

th a t any such singularities appear under an integral sign, where th eir contributions

rem ain finite. We do this by introducing the function

which leads to th e following non-singular Predholm integral equation of th e second

kind for th e function xjj (x) in th e interval x G [0,1],

i ; {x) = h { x ) V ^ J (x, () ijj (() (2.51)

T h e S o lu tio n o f th e Integral E q u ation o f th e Secon d K ind

We m ust now solve th e equation (2.51) for -0 (x) in th e interval x G [0,1]. We begin

by w riting (2.51) as

1 }

0 ( x ) = h (x ) + - / TV (x, 0 0 ( 0 (2.52)

7T J

0

where

N ( x , 0 = T - ^ M ( x , i ) . (2.53)

T he solution of (2.52) can be found using th e m ethod of successive substitution. See

Kondo (1991, page 43) . We su b stitu te for 0 (() on th e right-hand side of (2.52)

using th e expression (2.52) itself w ith x replaced by ( and ^ replaced by to obtain

I } r I }

0 ( x ) = h { x ) - \ ~ - N { x , ( ) h (() 4- - / ((, 6 ) 0 (&)

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1

11 =

h(x) + - j N { x , O h { O d ^ +

:^J J

N(x,^,)N{^uOi’ (Od^idt

0 0 0

(2.54)

S ubstituting successively for 'ip {^) in a similar m anner (n — 1) tim es gives

1 }

i ; ( x) = h{ x ) + - N { x , ^ ) h { ^ ) d ^ 7T J

0

1 1

+

lN{x,^,)N{^uî)h{Od^id^

0 0

1 ^ ^ ^

+ ^ / / / N 6 ) TV ( 6 , 6 ) TV ( 6 , 0 ( 0

0 0 0

11

1 + " ' + — y

J - J N (x,^i) •" N

(^n-l, 0

i’

(0

'

d^n-ld^-0 0 0

(2.55)

Assuming th e above p artial sum tends to a well defined lim it as n —)• oo we obtain

the solution

1 }

^l;(x) = h ( x ) + - N { x , ^ ) h ( ^ ) d ( 7T J

0

..

1 1

+ ^ / / i V ( a ; , f i ) i V ( ^ i , Ç ) A ( ^ ) d $ i d Ç

0 0

. . 1 1 1

+

^ / / 1 N(x,C,)Ni^u^2)N{^2,0hi0diid^2dî

0 0 0

i l l

1 +

" ' ^ J

y ' " y A

" (æ, (i) ' " N (^n-ij 0

h

(() •

d^n-id^

0 0 0

+ . . . (2.56)

(44)

2 .3 .4

T h e P r e s s u r e D iffe r e n c e S o lu tio n

We are finally in a position to w rite down th e solution for \p] {x). In th e interval

æ G [0,1] th e pressure difference takes th e form

[p] (^) = (^) (2.57)

where

1 }

'ip{x) = h( x ) + - N ( x , ^ ) h { ^ ) d ^ 7T J

0

^ 1 1

+

J J ^

^

—

0 0

1 1 1 1

+ —

J J ■

'

■ J

N {x, ^i ) ■ ’ • N (Cn-i, 0 h (() • • • d^n-id ^

H---0 0 0

(2.58)

In the above equation h {x) and N {x, ^) are defined to be

and

To com plete th e solution / (a:) is given by

1

7

f { x ) = - / l { ^ - x ) [t)] (() d i - {v) (a;) (2.61)

7T J —oo

where

^ = (^G2)

(45)

2 .4

N u m e r ic a l M e th o d s

2 .4 .1

T h e E v a lu a tio n o f t h e P r e s s u r e D iffe r e n c e

[p] {x)

T he solution as derived in th e previous section is far from simple in form and its

evaluation is not a straight-forw ard m atter. T he stru ctu re of th e solution is such

th a t we m ust consider its evaluation in stages. To s ta rt w ith we m ust determ ine

the function f (x) in the interval x G [0,1] using equation (2.61). We th en go

on to evaluate th e expressions (2.59) and (2.60) for h { x ) and th e kernel N {x, ^)

respectively. Finally, 'ip (x) is calculated using th e infinite sum (2.58). This completes

the solution in th e form (2.57)

Each of these stages presents its own difficulties, associated w ith th e accurate evalu

ation of a singular integral or an infinite sum, and we shall address each one in turn.

We will begin w ith th e last stage first, by considering an efficient way to calculate

the infinite sum appearing in equation (2.58).

2 .4 .2

T h e E v a lu a tio n o f

ip

(x)

We are faced w ith th e task of evaluating 'ip (x) in th e form of th e infinite sum (2.58)

which is the solution to the integral equation (2.52). T he ta sk is com pleted by using

the m ethod of successive approxim ations. We use th e form of (2.52) to define a

sequence of approxim ations to 'ip {x) which are calculated in an iterative manner.

The iteratio n is defined on th e interval x G [0,1] by

'ipo (x) = 0, (2.63)

1 r

ipn (x) = h{ x ) -h - N (T,

0

'ipn-i

(0

(2.64)

7T J

0

(46)

We m ust now show th a t this process does indeed yield a useful approxim ation to

the solution -0 (x). We do this in a sim ilar way to section (2.3.3), by substituting

for 'ipn-i iO above using equation (2.64) itself w ith x replaced by ( replaced by

^1, and n replaced by n — 1.

1 } r 1 }

0n (3^) = h{ x) - \ N {x, ^) h{^)-\--- / ((, &) ((i)

7T J 7T J

0 . 0 0

1 1 1

de

=

h{x) + ^ J N {x,^) h(^) dc+^ J j N {x,

e o

N

( e i , e ) V’n-2 ( e ) ^^ei^^e

0 0

(2.65)

By successively back-substituting in this way n tim es we ob tain th e following expres

sion for th e approxim ation to 0 (æ), 0 „ {x), in term s of th e initial approxim ation,

00 (a;).

1 }

'ipnix) = h{ x ) 4- - N { x , ^ ) h { ^ ) d ^ 7T J

0

1 1

+ N { x , ^ y ) N { ^ „ i ) h ( O d ^ i d i

0 0

1 1 1

+ ^ / /

J N(x,i,)N(i^,^,)N(i,,

0

h(i)diid^

2

di

0 0 0

1 1 1

+ ---— y

J

• "

J

N {x,^i) ' ■ ’ N (& -1 , ( ) 00 ( 0 - d^n-id^

0 0 0

(2.66)

In our case 0o (rr) = 0 and so th e last term disappears and we obtain

1 }

ijjnix) = h{ x ) + - / AT(T,() A (( ) d (

7T J

0

.. 1 1

+ 1 N { x , i , ) N { ^ u O h ( O d ^ i d i +

---0 0

1 1 1

+

^//•••/Af(^,ei)'"iv(en-2,e)A(e)dei---<ien-2de-0 ^//•••/Af(^,ei)'"iv(en-2,e)A(e)dei---<ien-2de-0 0